After selecting "Replay Battles" and going to username entry, I cannot continue any further. Pressing confirm on "OK" has no visible effect. And because you cannot back out of this screen, the only way to escape is to terminate the process.
After doing some web searches, I think the problem is that Everhood cannot connect to the internet (I'm guessing it's trying to update the leaderboards). I forbid Steam and its child processes from connecting to the internet when playing single-player games.
Ideally, I should be able to replay battles without leaderboards if I don't have internet access.